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Theorem plymul0or 22439
Description: Polynomial multiplication has no zero divisors. (Contributed by Mario Carneiro, 26-Jul-2014.)
Assertion
Ref Expression
plymul0or  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( F  oF  x.  G
)  =  0p  <-> 
( F  =  0p  \/  G  =  0p ) ) )

Proof of Theorem plymul0or
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 dgrcl 22393 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
2 dgrcl 22393 . . . . . . 7  |-  ( G  e.  (Poly `  S
)  ->  (deg `  G
)  e.  NN0 )
3 nn0addcl 10831 . . . . . . 7  |-  ( ( (deg `  F )  e.  NN0  /\  (deg `  G )  e.  NN0 )  ->  ( (deg `  F )  +  (deg
`  G ) )  e.  NN0 )
41, 2, 3syl2an 477 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( (deg `  F )  +  (deg
`  G ) )  e.  NN0 )
5 c0ex 9590 . . . . . . 7  |-  0  e.  _V
65fvconst2 6116 . . . . . 6  |-  ( ( (deg `  F )  +  (deg `  G )
)  e.  NN0  ->  ( ( NN0  X.  {
0 } ) `  ( (deg `  F )  +  (deg `  G )
) )  =  0 )
74, 6syl 16 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( NN0  X.  { 0 } ) `  ( (deg
`  F )  +  (deg `  G )
) )  =  0 )
8 fveq2 5866 . . . . . . . 8  |-  ( ( F  oF  x.  G )  =  0p  ->  (coeff `  ( F  oF  x.  G
) )  =  (coeff `  0p ) )
9 coe0 22415 . . . . . . . 8  |-  (coeff ` 
0p )  =  ( NN0  X.  {
0 } )
108, 9syl6eq 2524 . . . . . . 7  |-  ( ( F  oF  x.  G )  =  0p  ->  (coeff `  ( F  oF  x.  G
) )  =  ( NN0  X.  { 0 } ) )
1110fveq1d 5868 . . . . . 6  |-  ( ( F  oF  x.  G )  =  0p  ->  ( (coeff `  ( F  oF  x.  G ) ) `
 ( (deg `  F )  +  (deg
`  G ) ) )  =  ( ( NN0  X.  { 0 } ) `  (
(deg `  F )  +  (deg `  G )
) ) )
1211eqeq1d 2469 . . . . 5  |-  ( ( F  oF  x.  G )  =  0p  ->  ( (
(coeff `  ( F  oF  x.  G
) ) `  (
(deg `  F )  +  (deg `  G )
) )  =  0  <-> 
( ( NN0  X.  { 0 } ) `
 ( (deg `  F )  +  (deg
`  G ) ) )  =  0 ) )
137, 12syl5ibrcom 222 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( F  oF  x.  G
)  =  0p  ->  ( (coeff `  ( F  oF  x.  G ) ) `  ( (deg `  F )  +  (deg `  G )
) )  =  0 ) )
14 eqid 2467 . . . . . . 7  |-  (coeff `  F )  =  (coeff `  F )
15 eqid 2467 . . . . . . 7  |-  (coeff `  G )  =  (coeff `  G )
16 eqid 2467 . . . . . . 7  |-  (deg `  F )  =  (deg
`  F )
17 eqid 2467 . . . . . . 7  |-  (deg `  G )  =  (deg
`  G )
1814, 15, 16, 17coemulhi 22413 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( (coeff `  ( F  oF  x.  G ) ) `
 ( (deg `  F )  +  (deg
`  G ) ) )  =  ( ( (coeff `  F ) `  (deg `  F )
)  x.  ( (coeff `  G ) `  (deg `  G ) ) ) )
1918eqeq1d 2469 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( (
(coeff `  ( F  oF  x.  G
) ) `  (
(deg `  F )  +  (deg `  G )
) )  =  0  <-> 
( ( (coeff `  F ) `  (deg `  F ) )  x.  ( (coeff `  G
) `  (deg `  G
) ) )  =  0 ) )
2014coef3 22392 . . . . . . . 8  |-  ( F  e.  (Poly `  S
)  ->  (coeff `  F
) : NN0 --> CC )
2120adantr 465 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (coeff `  F
) : NN0 --> CC )
221adantr 465 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (deg `  F
)  e.  NN0 )
2321, 22ffvelrnd 6022 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( (coeff `  F ) `  (deg `  F ) )  e.  CC )
2415coef3 22392 . . . . . . . 8  |-  ( G  e.  (Poly `  S
)  ->  (coeff `  G
) : NN0 --> CC )
2524adantl 466 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (coeff `  G
) : NN0 --> CC )
262adantl 466 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (deg `  G
)  e.  NN0 )
2725, 26ffvelrnd 6022 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( (coeff `  G ) `  (deg `  G ) )  e.  CC )
2823, 27mul0ord 10199 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( (
( (coeff `  F
) `  (deg `  F
) )  x.  (
(coeff `  G ) `  (deg `  G )
) )  =  0  <-> 
( ( (coeff `  F ) `  (deg `  F ) )  =  0  \/  ( (coeff `  G ) `  (deg `  G ) )  =  0 ) ) )
2919, 28bitrd 253 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( (
(coeff `  ( F  oF  x.  G
) ) `  (
(deg `  F )  +  (deg `  G )
) )  =  0  <-> 
( ( (coeff `  F ) `  (deg `  F ) )  =  0  \/  ( (coeff `  G ) `  (deg `  G ) )  =  0 ) ) )
3013, 29sylibd 214 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( F  oF  x.  G
)  =  0p  ->  ( ( (coeff `  F ) `  (deg `  F ) )  =  0  \/  ( (coeff `  G ) `  (deg `  G ) )  =  0 ) ) )
3116, 14dgreq0 22424 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  ( F  =  0p  <->  ( (coeff `  F ) `  (deg `  F ) )  =  0 ) )
3231adantr 465 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( F  =  0p  <->  ( (coeff `  F ) `  (deg `  F ) )  =  0 ) )
3317, 15dgreq0 22424 . . . . 5  |-  ( G  e.  (Poly `  S
)  ->  ( G  =  0p  <->  ( (coeff `  G ) `  (deg `  G ) )  =  0 ) )
3433adantl 466 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( G  =  0p  <->  ( (coeff `  G ) `  (deg `  G ) )  =  0 ) )
3532, 34orbi12d 709 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( F  =  0p  \/  G  =  0p )  <->  ( (
(coeff `  F ) `  (deg `  F )
)  =  0  \/  ( (coeff `  G
) `  (deg `  G
) )  =  0 ) ) )
3630, 35sylibrd 234 . 2  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( F  oF  x.  G
)  =  0p  ->  ( F  =  0p  \/  G  =  0p ) ) )
37 cnex 9573 . . . . . . 7  |-  CC  e.  _V
3837a1i 11 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  CC  e.  _V )
39 plyf 22358 . . . . . . 7  |-  ( G  e.  (Poly `  S
)  ->  G : CC
--> CC )
4039adantl 466 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  G : CC
--> CC )
41 0cnd 9589 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  0  e.  CC )
42 mul02 9757 . . . . . . 7  |-  ( x  e.  CC  ->  (
0  x.  x )  =  0 )
4342adantl 466 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  x  e.  CC )  ->  ( 0  x.  x )  =  0 )
4438, 40, 41, 41, 43caofid2 6555 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( CC  X.  { 0 } )  oF  x.  G )  =  ( CC  X.  { 0 } ) )
45 id 22 . . . . . . . 8  |-  ( F  =  0p  ->  F  =  0p
)
46 df-0p 21840 . . . . . . . 8  |-  0p  =  ( CC  X.  { 0 } )
4745, 46syl6eq 2524 . . . . . . 7  |-  ( F  =  0p  ->  F  =  ( CC  X.  { 0 } ) )
4847oveq1d 6299 . . . . . 6  |-  ( F  =  0p  -> 
( F  oF  x.  G )  =  ( ( CC  X.  { 0 } )  oF  x.  G
) )
4948eqeq1d 2469 . . . . 5  |-  ( F  =  0p  -> 
( ( F  oF  x.  G )  =  ( CC  X.  { 0 } )  <-> 
( ( CC  X.  { 0 } )  oF  x.  G
)  =  ( CC 
X.  { 0 } ) ) )
5044, 49syl5ibrcom 222 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( F  =  0p  -> 
( F  oF  x.  G )  =  ( CC  X.  {
0 } ) ) )
51 plyf 22358 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )
5251adantr 465 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  F : CC
--> CC )
53 mul01 9758 . . . . . . 7  |-  ( x  e.  CC  ->  (
x  x.  0 )  =  0 )
5453adantl 466 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  x  e.  CC )  ->  ( x  x.  0 )  =  0 )
5538, 52, 41, 41, 54caofid1 6554 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( F  oF  x.  ( CC  X.  { 0 } ) )  =  ( CC  X.  { 0 } ) )
56 id 22 . . . . . . . 8  |-  ( G  =  0p  ->  G  =  0p
)
5756, 46syl6eq 2524 . . . . . . 7  |-  ( G  =  0p  ->  G  =  ( CC  X.  { 0 } ) )
5857oveq2d 6300 . . . . . 6  |-  ( G  =  0p  -> 
( F  oF  x.  G )  =  ( F  oF  x.  ( CC  X.  { 0 } ) ) )
5958eqeq1d 2469 . . . . 5  |-  ( G  =  0p  -> 
( ( F  oF  x.  G )  =  ( CC  X.  { 0 } )  <-> 
( F  oF  x.  ( CC  X.  { 0 } ) )  =  ( CC 
X.  { 0 } ) ) )
6055, 59syl5ibrcom 222 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( G  =  0p  -> 
( F  oF  x.  G )  =  ( CC  X.  {
0 } ) ) )
6150, 60jaod 380 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( F  =  0p  \/  G  =  0p )  ->  ( F  oF  x.  G
)  =  ( CC 
X.  { 0 } ) ) )
6246eqeq2i 2485 . . 3  |-  ( ( F  oF  x.  G )  =  0p  <->  ( F  oF  x.  G )  =  ( CC  X.  { 0 } ) )
6361, 62syl6ibr 227 . 2  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( F  =  0p  \/  G  =  0p )  ->  ( F  oF  x.  G
)  =  0p ) )
6436, 63impbid 191 1  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( F  oF  x.  G
)  =  0p  <-> 
( F  =  0p  \/  G  =  0p ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3113   {csn 4027    X. cxp 4997   -->wf 5584   ` cfv 5588  (class class class)co 6284    oFcof 6522   CCcc 9490   0cc0 9492    + caddc 9495    x. cmul 9497   NN0cn0 10795   0pc0p 21839  Polycply 22344  coeffccoe 22346  degcdgr 22347
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570  ax-addf 9571
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-of 6524  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-sup 7901  df-oi 7935  df-card 8320  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-n0 10796  df-z 10865  df-uz 11083  df-rp 11221  df-fz 11673  df-fzo 11793  df-fl 11897  df-seq 12076  df-exp 12135  df-hash 12374  df-cj 12895  df-re 12896  df-im 12897  df-sqrt 13031  df-abs 13032  df-clim 13274  df-rlim 13275  df-sum 13472  df-0p 21840  df-ply 22348  df-coe 22350  df-dgr 22351
This theorem is referenced by:  plydiveu  22456  quotcan  22467  vieta1lem1  22468  vieta1lem2  22469
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